Subgroups — #26106328

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	2	yes	maximal
2	#20241269	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	C₄×#10625329	—	2	4	no	—
6	C₄×#10625329	C₄	1	8	yes	complemented
7	C₂×#13093935	C₂²	1	4	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	C₂×#117×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#384	C₄	1	2	yes	—
12	C₄₇×#382	C₂²	1	2	yes	—
13	C₄₇×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
15	C₂×#10625329	—	2	16	no	—
16	C₂×#10625329	—	2	8	no	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	#13093935	—	2	4	no	—
19	#13093935	C₂×C₄	1	8	yes	—
20	C₄×C₄₇⋊C₈	—	2	8	no	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	D₄	1	2	yes	—
23	#117×C₄₇	—	4	4	no	—
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#95×C₄₇	—	2	4	no	—
35	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	8	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₄⋊C₄×C₄₇	—	4	4	no	—
49	C₄⋊C₄×C₄₇	—	4	8	no	—
50	C₂²⋊C₄×C₄₇	—	4	4	no	—
51	C₄²×C₄₇	—	4	4	no	—
52	C₄²×C₄₇	C₄²	1	2	yes	—
53	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
54	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	#10625329	—	2	32	no	—
60	C₂×C₄₇⋊C₈	—	2	16	no	—
61	C₂×C₄₇⋊C₈	—	2	8	no	—
62	C₂³×C₄₇	—	2	1	no	—
63	C₂³×C₄₇	—	2	1	no	—
64	C₂³×C₄₇	—	2	4	no	—
65	C₂³×C₄₇	#95	1	2	yes	—
66	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	8	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	4	4	no	—
75	C₂×C₄×C₄₇	—	4	8	no	—
76	C₂×C₄×C₄₇	#95	1	4	yes	—
77	C₂×C₄×C₄₇	#95	1	2	yes	—
78	C₂×C₄×C₄₇	#100	1	2	yes	—
79	C₂×C₄×C₄₇	#101	1	2	yes	—
80	C₂×C₄×C₄₇	#104	1	8	yes	—
81	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄₇⋊C₈	—	2	16	no	—
86	#8028	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	4	no	—
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	—	4	2	no	—
90	C₂²×C₄₇	—	4	2	no	—
91	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
92	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
94	C₂²×C₄₇	C₂×#104	1	4	yes	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	2	1	no	—
99	C₄×C₄₇	—	8	4	no	—
100	C₄×C₄₇	#339	1	8	yes	—
101	C₄×C₄₇	#343	1	4	yes	—
102	C₂×C₄×M₁₆	—	47	2	no	—
103	#2285	D₄₇	1	1	yes	characteristic
104	#1295	—	47	4	no	—
105	C₂×C₄₇	—	4	2	no	—
106	C₂×C₄₇	#1525	1	2	yes	—
107	C₂×C₄₇	#1526	1	2	yes	—
108	C₂×C₄₇	C₂×#339	1	2	yes	—
109	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
110	C₂²×M₁₆	—	94	2	no	—
111	C₂×#117	Dic₄₇	1	2	yes	—
112	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
113	C₄×M₁₆	—	47	8	no	—
114	C₄×M₁₆	—	94	4	no	—
115	C₂×#97	—	47	4	no	—
116	C₂×C₄×C₈	—	94	2	no	—
117	#384	Dic₄₇	1	2	yes	—
118	#382	C₂×D₄₇	1	2	yes	—
119	#380	—	2	2	no	—
120	C₄×C₄⋊C₄	—	2	2	no	—
121	C₄₇	#8028	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
123	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
124	C₂×M₁₆	—	94	16	no	—
125	C₂×M₁₆	—	94	8	no	—
126	C₂²×C₈	—	94	4	no	—
127	#117	—	4	4	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
130	C₂×C₂²⋊C₄	—	2	2	no	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	Dic₉₄	1	2	yes	—
133	C₂×C₄²	C₄×D₄₇	1	4	yes	—
134	C₂×C₄²	D₁₈₈	1	2	yes	—
135	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
137	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
138	#97	—	47	8	no	—
139	#97	—	94	4	no	—
140	C₄×C₈	—	94	8	no	—
141	#95	—	2	4	no	—
142	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
148	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
149	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
150	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
151	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
152	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
153	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
155	M₁₆	—	94	32	no	—
156	C₂×C₈	—	94	16	no	—
157	C₂×C₈	—	94	8	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₄×Dic₄₇	1	2	yes	—
163	C₄²	Dic₄₇⋊C₄	1	4	yes	—
164	C₄²	C₄⋊Dic₄₇	1	2	yes	—
165	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
166	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
167	C₄²	C₂×C₄×D₄₇	1	2	yes	—
168	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	4	no	—
172	C₂³	C₄₇⋊#95	1	2	yes	—
173	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	C₄₇⋊#95	1	4	yes	—
184	C₂×C₄	C₄₇⋊#95	1	2	yes	—
185	C₂×C₄	#13093941	1	2	yes	—
186	C₂×C₄	#13093943	1	2	yes	—
187	C₂×C₄	#13093951	1	8	yes	—
188	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
189	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
190	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
192	C₈	—	94	16	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
198	C₂²	C₂×#13093941	1	1	yes	characteristic
199	C₂²	C₂×#13093943	1	1	yes	characteristic
200	C₂²	C₂×#13093951	1	4	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#15688882	1	8	yes	—
207	C₄	#15688892	1	4	yes	—
208	C₂	—	4	2	no	—
209	C₂	#20241980	1	2	yes	—
210	C₂	#20241983	1	2	yes	—
211	C₂	C₂×#15688882	1	2	yes	—
212	C₂	C₂×#15688892	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#8028
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20241269	C₂	semidirect
6. C₄×#10625329	C₄	semidirect
121. C₄₇	#8028	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#10625329	2
#20241269	2
C₄₇×#2285	2
#8028	47